1. Searching by Constraint & Searching by Evolution
CMSC 25000
Artificial Intelligence
January 20, 2005

2. Agenda Constraint Propagation: Review
Constraint Propagation Mechanisms
CSP as search
Forward-checking
Back-jumping
Iterative refinement: min-conflict
Summary

3. Constraint Satisfaction Problems
Very general: Model wide range of tasks
Key components:
Variables: Take on a value
Domains: Values that can be assigned to vars
Constraints: Restrictions on assignments
Constraints are HARD
Not preferences: Must be observed
E.g. Can’t schedule two classes: same room, same time

4. Representational Advantages
Simple goal test:
Complete, consistent assignment
Complete: All variables have a value
Consistent: No constraints violates
Maximum depth?
Number of variables
Search order?
Commutative, reduces branching
Strategy: Assign value to one variable at a time

5. Constraint Satisfaction Questions
Can we rule out possible search paths based on current values and constraints?
How should we pick the next variable to assign?
Which value should we assign next?
Can we exploit problem structure for efficiency?

6. CSP as Search
Constraint Satisfaction Problem (CSP) is a restricted class of search problem
State: Set of variables & assignment info
Initial State: No variables assigned
Goal state: Set of assignments consistent with constraints
Operators: Assignment of value to variable

7. CSP as Search Depth: Number of Variables Branching Factor: |Domain|
Leaves: Complete Assignments
Blind search strategy:
Bounded depth -> Depth-first strategy
Backtracking:
Recover from dead ends
Find satisfying assignment

8. Constraint Propagation Mechanism
Arc consistency
Arc V1->V2 is arc consistent if for all x in D1, there exists y in D2 such that (x,y) is allowed
Delete disallowed x’s to achieve arc consistency

9. Arc Consistency Example
Graph Coloring:
Variables: Nodes
Domain: Colors (R,G,B)
Constraint: If E(a,b), then C(a) != C(b)
Initial assignments
R,G,B X
R,B B Y Z

10. Arc Consistency Example
Arc Rm Value
Assignments:
X = G
Y = R
Z = B
X -> Y None
X -> Z X=B
======>
Y-> Z Y=B
X -> Y X=R
X -> Z None
Y -> Z None
R,G,B X
R,B B Y Z

11. Limitations of Arc Consistency
Arc consistent:
No legal assignment
>1 legal assignment
G,B X
G,B
G,B Y Z
R,G X
G,B
G,B Y Z

12. Constraint Graph
R,G,B X Y Z
R,B
R,B

13. CSP as Search X=R X=G X=B Y=R Y=B Y=R Y=B Y=R Y=B No! No! Z=B Z=R Z=B
Yes!
No!
No!
Yes!
No!
No!

14. Backtracking Issues CSP as Search Depth-first search
Maximum depth = # of variables
Continue until (partial/complete) assignment
Consistent: return result
Violates constraint -> backtrack to previous assignment
Wasted effort
Explore branches with no possible legal assignment

15. Backtracking with Forward Checking
Augment DFS with backtracking
Add some constraint propagation
Propagate constraints
Remove assignments which violate constraint
Only propagate when domain = 1
“Forward checking”
Limit constraints to test

16. Backtracking+Forward Checking
X=R
X=G
X=B
Y=B
Y=R
Y=B
Y=R
Z=B
Z=R
No!
No!
Yes!
Yes!

17. Heuristic CSP Improving Backtracking Standard backtracking
Back up to last assignment
Back-jumping
Back up to last assignment that reduced current domain
Change assignment that led to dead end

18. Heuristic backtracking: Back-jumping

19. Heuristic Backtracking: Backjumping

20. Heuristic Backtracking: Backjumping
Dead end! Why? C5

21. Back-jumping Previous assignment reduced domain
C3 = B
Changes to intermediate assignment can’t affect dead end
Backtrack up to C3
Avoid wasted work - alternatives at C4
In general, forward checking more effective

22. Heuristic CSP: Dynamic Ordering
Question: What to explore next
Current solution:
Static: Next in fixed order
Lexicographic, leftmost..
Random
Alternative solution:
Dynamic: Select “best” in current state

23. Question How would you pick a variable to assign?
How would you pick a value to assign?

24. Dynamic Ordering Heuristic CSP
Most constrained variable:
Pick variable with smallest current domain
Least constraining value:
Pick value that removes fewest values from domains of other variables
Improve chance of finding SOME assignment
Can increase feasible size of CSP problem

25. Dynamic Ordering C1 C3 C5 C2 C4

26. Dynamic Ordering with FC

27. Incremental Repair Start with initial complete assignment
Use greedy approach
Probably invalid - I.e. violates some constraints
Incrementally convert to valid solution
Use heuristic to replace value that violates
“min-conflict” strategy:
Change value to result in fewest constraint violations
Break ties randomly
Incorporate in local or backtracking hill-climber

28. Incremental Repair Q2 Q4 5 conflicts Q1 Q3 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4

29. Question
How would we apply iterative repair to Traveling Salesman Problem?

30. Min-Conflict Effectiveness
N-queens: Given initial random assignment, can solve in ~ O(n)
For n < 10^7
GSAT (satisfiability)
Best (near linear in practice) solution uses min-conflict-type hill-climbing strategy
Adds randomization to escape local min
~Linear seems true for most CSPs
Except for some range of ratios of constraints to variables
Avoids storage of assignment history (for BT)

31. Constraint Propagation Method
For each variable,
Get all values for that variable
Remove all values that conflict with ALL adjacent
A:For each neighbor, remove all values that conflict with ALL adjacent
Repeat A as long as some label set is reduced

32. Constraint Propagation Example
Image interpretation:
From a 2-D line drawing, automatically determine for each line, if it forms a concave, convex or boundary surface
Segment image into objects
Simplifying assumptions:
World of polyhedra
No shadows, no cracks

33. CSP Example: Line Labeling
3 Line Labels:
Boundary line: regions do not abut: >
Arrow direction: right hand side walk
Interior line: regions do abut
Concave edge line: _
Convex edge line : +
Junction labels:
Where line labels meet

34. CSP Example: Line Labeling
Simplifying (initial) restrictions:
No shadows, cracks
Three-faced vertexes
General position:
small changes in view-> no change in junction type
4^2 labels for 2 line junction: L
4^3; 3-line junction : Fork, Arrow, T
Total: 208 junction labelings

35. CSP Example: Line Labeling
Key observation: Not all 208 realizable
How many? 18 physically realizable
All Junctions
L junctions Fork Junctions T junctions Arrow junctions + + _ + _ + _ + _ + + + _ _ _ _ + _ _ _ _ +

36. CSP Example: Line Labeling
Label boundaries clockwise
Label arrow junctions + + +

37. CSP Example: Line Labeling
Label boundaries clockwise

38. CSP Example: Line Labeling
Label fork junctions with +’s + + + + + + + + + + + + + + +

39. CSP Example: Line Labeling
Label arrow junctions with -’s + + + _ + + _ + + + _ _ _ + _ _ + + + _ + + _ + + + +

40. Waltz’s Line Labeling For each junction in image,
Get all labels for that junction type
Remove all labels that conflict with ALL adjacent
A:For each neighbor, remove all labels that conflict with ALL
Repeat A as long as some label set is reduced

41. Waltz Propagation Example+ + C D + _ X B X _ + A X

42. Waltz’s Line Labeling Full version: Physically realizable junctions
Removes constraints on images
Adds special shadow and crack labels
Includes all possible junctions
Not just 3 face
Physically realizable junctions
Still small percentage of all junction labels
O(n) : n = # lines in drawing

43. Agenda Motivation: Genetic Algorithms Conclusions Evolving a solution
Modeling search as evolution
Mutation
Crossover
Survival of the fittest
Survival of the most diverse
Conclusions

44. Motivation: Evolution
Evolution through natural selection
Individuals pass on traits to offspring
Individuals have different traits
Fittest individuals survive to produce more offspring
Over time, variation can accumulate
Leading to new species

45. Motivation: Evolution
Passing on traits to offspring
Chromosomes carry genes for different traits
Usually chromosomes paired - one from each parent
Chromosomes are duplicated before mating
Crossover mixes genetic material from chromosomes
Each parent produces one single chromosome cell
Mating joins cells
Mutation: error in duplication -> different gene

46. Evolution Variation: Arises from crossover & mutation
Crossover: Produces new gene combinations
Mutation: Produces new genes
Different traits lead to different fitnesses

47. Simulated Evolution Evolving a solution
Begin with population of individuals
Individuals = candidate solutions ~chromosomes
Produce offspring with variation
Mutation: change features
Crossover: exchange features between individuals
Apply natural selection
Select “best” individuals to go on to next generation
Continue until satisfied with solution

48. Genetic Algorithms Applications
Search parameter space for optimal assignment
Not guaranteed to find optimal, but can approach
Classic optimization problems:
E.g. Traveling Salesman Problem
Program design (“Genetic Programming”)
Aircraft carrier landings

49. Question
How can we encode TSP as GA?
Demo

50. Genetic Algorithm Example
Cookie recipes (Winston, AI, 1993)
As evolving populations
Individual = batch of cookies
Quality: 0-9
Chromosomes = 2 genes: 1 chromosome each
Flour Quantity, Sugar Quantity: 1-9
Mutation:
Randomly select Flour/Sugar: +/- 1 [1-9]
Crossover:
Split 2 chromosomes & rejoin; keeping both

51. Mutation & Crossover Mutation: 1 1 1 1 2 1 1 2 2 2 1 3 Crossover: 2 2

52. Fitness Natural selection: Most fit survive
Fitness= Probability of survival to next gen
Question: How do we measure fitness?
“Standard method”: Relate fitness to quality
:0-1; :1-9:
Chromosome Quality Fitness
1 4
3 1
1 2
1 1 4 3 2 1
0.4
0.3
0.2
0.1

53. Genetic Algorithms Procedure
Create an initial population (1 chromosome)
Mutate 1+ genes in 1+ chromosomes
Produce one offspring for each chromosome
Mate 1+ pairs of chromosomes with crossover
Add mutated & offspring chromosomes to pop
Create new population
Best + randomly selected (biased by fitness)

54. GA Design Issues Genetic design: Population: How many chromosomes?
Identify sets of features = genes; Constraints?
Population: How many chromosomes?
Too few => inbreeding; Too many=>too slow
Mutation: How frequent?
Too few=>slow change; Too many=> wild
Crossover: Allowed? How selected?
Duplicates?

55. GA Design: Basic Cookie GA
Genetic design:
Identify sets of features: 2 genes: flour+sugar;1-9
Population: How many chromosomes?
1 initial, 4 max
Mutation: How frequent?
1 gene randomly selected, randomly mutated
Crossover: Allowed? No
Duplicates? No
Survival: Standard method

56. Example Mutation of 2 Chromosome Quality 1 4 4 Generation 0: 2 2 3
Generation 0:
Chromosome Quality
Generation 1:
Chromosome Quality
Generation 3:
Chromosome Quality
Generation 2:
Chromosome Quality

57. Basic Cookie GA Results
Results are for 1000 random trials
Initial state: 1 1-1, quality 1 chromosome
On average, reaches max quality (9) in 16 generations
Best: max quality in 8 generations
Conclusion:
Low dimensionality search
Successful even without crossover

58. Adding Crossover Genetic design: Population: How many chromosomes?
Identify sets of features: 2 genes: flour+sugar;1-9
Population: How many chromosomes?
1 initial, 4 max
Mutation: How frequent?
1 gene randomly selected, randomly mutated
Crossover: Allowed? Yes, select random mates; cross at middle
Duplicates? No
Survival: Standard method

59. Basic Cookie GA+Crossover Results
Results are for 1000 random trials
Initial state: 1 1-1, quality 1 chromosome
On average, reaches max quality (9) in 14 generations
Conclusion:
Faster with crossover: combine good in each gene
Key: Global max achievable by maximizing each dimension independently - reduce dimensionality

60. Solving the Moat Problem
No single step mutation can reach optimal values using standard fitness (quality=0 => probability=0)
Solution A:
Crossover can combine fit parents in EACH gene
However, still slow: 155 generations on average 1 2 3 4 5 4 3 2 1 2 2 3 3 4 7 8 7 4 5 8 9 8 5 4 7 8 7 4 3 3 2 2 1 2 3 4 5 4 3 2 1

61. Questions How can we avoid the 0 quality problem?
How can we avoid local maxima?

62. Rethinking Fitness Goal: Explicit bias to best Solution: Rank method
Remove implicit biases based on quality scale
Solution: Rank method
Ignore actual quality values except for ranking
Step 1: Rank candidates by quality
Step 2: Probability of selecting ith candidate, given that i-1 candidate not selected, is constant p.
Step 2b: Last candidate is selected if no other has been
Step 3: Select candidates using the probabilities

63. Rank Method Chromosome Quality Rank Std. Fitness Rank Fitness 1 4 1 3
1 2
5 2
7 5 4 3 2 1 1 2 3 4 5
0.4
0.3
0.2
0.1
0.0
0.667
0.222
0.074
0.025
0.012
Results: Average over 1000 random runs on Moat problem
- 75 Generations (vs 155 for standard method)
No 0 probability entries: Based on rank not absolute quality

64. Diversity
Diversity:
Degree to which chromosomes exhibit different genes
Rank & Standard methods look only at quality
Need diversity: escape local min, variety for crossover
“As good to be different as to be fit”

65. Rank-Space Method Combines diversity and quality in fitness
Diversity measure:
Sum of inverse squared distances in genes
Diversity rank: Avoids inadvertent bias
Rank-space:
Sort on sum of diversity AND quality ranks
Best: lower left: high diversity & quality

66. Rank-Space Method W.r.t. highest ranked 5-1
Chromosome Q D D Rank Q Rank Comb Rank R-S Fitness
1 4
3 1
1 2
1 1
7 5 4 3 2 1
0.04
0.25
0.059
0.062
0.05 1 5 3 4 2 1 2 3 4 5 1 4 2 5 3
0.667
0.025
0.222
0.012
0.074
Diversity rank breaks ties
After select others, sum distances to both
Results: Average (Moat) 15 generations

67. GA’s and Local Maxima Quality metrics only: Quality + Diversity:
Susceptible to local max problems
Quality + Diversity:
Can populate all local maxima
Including global max
Key: Population must be large enough

68. Genetic Algorithms Evolution mechanisms as search technique
Produce offspring with variation
Mutation, Crossover
Select “fittest” to continue to next generation
Fitness: Probability of survival
Standard: Quality values only
Rank: Quality rank only
Rank-space: Rank of sum of quality & diversity ranks
Large population can be robust to local max